Field generators in two variables

Very good and very bad field generators

“…Let us be precise about the use of language (regarding dicriticals or some equivalent concept) in [ACN96], [CN05], [Rus75] and [Rus77], since we are going to refer to those papers.…”

“…Papers [Rus75] and [Rus77] simply speak of points at infinity, instead of dicriticals. Let us explain this.…”

“…Then L/K is a function field in one variable. Let C F be the complete regular curve over K whose function field is L. Let P be a closed point of C F and (O P , m P ) the local ring of C F at P ; by definition of the degree of a point on a variety, deg [Rus75] and [Rus77]) that P is a "point of C F at infinity" (or a "place of C F at infinity"). As is well known, the set of closed points of C F can be identified with V(L/K) via the bijection P → O P ; then the set V ∞ (F, A) of dicriticals is precisely the set of points of C F at infinity and the degrees of the dicriticals are the degrees of the points.…”

“…. , n}, let Λ (i) be the strict (or proper) transform of Λ on S i (defined in [Rus75], just before 2.4); note that Λ (0) = Λ.…”

“…Field generators are studied in [Jan74], [Rus75], [Rus77] and [CN05]. The first example of bad field generator was given in [Jan74], and more examples were given in [Rus77] and [CN05].…”